Passive radiator speaker design – Box calculation example
What is a passive radiator ?
A passive radiator speaker design involves another speaker, but with no motor assembly. The magnet and voice coil are gone, but the cone and everything you can see from the front are all there. The passive radiator subwoofer design is very similar to the bass reflex design, only instead of the vent or port, you have the passive bass radiator. Since they are very similar in design principle, they follow similar methodology and performance characteristics. However, certain differences do apply, some are good, some are not so desirable.
Advantages over bass-reflex :
- No more vent non-linearities (resonant pipe sounds).
- Air turbulence noise is no longer an issue, when air rapidly escapes the pipe at high volumes.
- No more high frequencies reflected out the port.
- Space efficient. Sometimes large ports are demanded which will make the enclosure that much bigger.
- Passive radiator speaker design is simpler and with fewer alignments.
Disadvantages over bass-reflex :
- Steeper roll-off.
- Less transient stability.
- Slightly higher cut-off frequency (narrower frequency bandwidth).
- Greater overall box losses (QL).
Passive radiators operate in conjunction with the active driver at low frequencies, sharing the acoustic load and reduce the excursion of the driver. Working similarly to a vent, the passive radiator will only add as much as they subtract. This means they will have the same advantages as the bass reflex port : higher power handling and lower distortion.
How does the passive bass radiator compare to the alternatives?
As we mentioned before, the response for the passive radiator speaker design is closely related to that of the bass reflex. The one thing that is unique to the passive radiator is the notch at the resonant frequency of the passive speaker. Usually, the resonance is located at the point where the response start to roll-off (f0), but for the passive bass radiator, this point is below the system’s cut-off frequency. This notch increases the slope of the driver low-frequency roll-off, and degrades the transient response.
As you can see from the graph, sealed has the smoothest roll-off, followed by bass-reflex with a steeper roll-off but with a broader frequency response. Passive radiator has the middle ground when it comes to frequency response bandwidth, but has the steepest roll-off slope. Notice the notch in the response curve, that we talked about earlier. The vented enclosure will have more low end than the PR box, but the passive radiator design will not suffer from pipe resonances and standing waves transmission problems.
Which speaker is best for a passive radiator speaker design ?
If compared to the closely related bass-reflex design, the passive radiator (PR) design is more picky when it comes to choosing you speaker. When it comes to passive radiator alignments, they are fewer options, compared to the ported design. Please read the article regarding the bass reflex alignments before continuing. When it comes to PR, you are restricted to 3rd order Quasi Butterworth (QB3), 4th order Butterworth (B4) and 4th order Chebyshev (C4). This is because any driver with a Qts higher than 0.5 will produce a C4 response with a high amount of ripple (large peak near cut-off).
If you want to get a flat response you will need a Qts that is in between 0.2 and 0.35 values to get a nice QB3 design. Usually when you making a passive radiator speaker design, you will probably use a PR that is the same size as the active speaker. This means it has the same cone area, weight and compliance. However, if the radiator is made more compliant than the active speaker, higher Qts alignments will be more acceptable and are similar to the bass-reflex conterparts, with acceptable transient response.
Parameters used for designing the box
Before you get started you will need the following parameters, by either measuring them or by getting them from the manufacturer :
- fs – resonant frequency of the active speaker.
- Qts – Total Q.
- Vas – equivalent compliance.
- Sd – effective area of the cone.
- Vd – volume displaced by the speaker.
- δ (delta) – the compliance ratio of the passive radiator.
Using at home measurements to calculate δ is a challenging and tedious task. Hopefully the passive radiator manufacturer has provide enough information for you to calculate it without any hassle.
Please review the following parameters :
- Cab – the acoustic compliance of the enclosure.
- Cap – the acoustic compliance of the PR.
- Vb – Volume of the enclosure.
- Vas – Equivalent compliance of the driver.
δ = Cap / Cab
α = Vas / Vb
The following list of alignments of passive radiator speaker design takes into consideration that QL = 7 and δ = α. So to spare you any tedious calculations and complications, when you choose your passive radiator and active speaker, make sure they match each other. If they have the same Vas and Cms you are good to go and then you can use the table below. By doing this you don’t have to calculate δ.
Now let’s use a real world example of active-passive speaker combo, to make an enclosure following the alignments table. For our example, we will use two speakers from Scanspeak. Here are the links to the technical information :
Passive radiator calculator – enclosure design
As you can see, both Cms and Vas are the same for the two speakers. This means that δ = α and we can safely use the information in the alignments table. Qts = 0.31, so we use the information and that line to calculate the following parameters :
- Volume of the box : Vb = Vas / α = 95 / 2.78 = 34.17 L.
- Tuning frequency : fB = H * fs = 1.51 * 21 = 32 Hz.
- –3 db point : f3 = (f3/fs) * fs = 1.59 * 21 = 34 Hz.
- PR displacement : Vpr = (Vpr/Vd) * Vd = 2.35 * 0.00044 = 0.001034 m3
- Vd = Sd * Xmax = 0.0352 m2 * 0.0125 m = 0.00044 m3
Of course you don’t have to follow these alignments and make the enclosure to your desired response and tuning frequency. To alter the tuning frequency, you do it similar to the bass reflex enclosure. Instead of modifying the length of the port you modify the mass of the passive radiator. Adding clay or swaping the dust cap for a different material. Here is a Seas passive radiator, which shows how adding mass to the cone changes resonant frequency.
Conclusion on the passive radiator subwoofer design
The passive radiator is intended to be used as an alternative to the ported design. Pro’s and con’s have already been discussed and you should read them carefully to make a good decision if the passive radiator speaker design is for you. Also take note of the price. The passive speaker will be cheaper than the active one because they are similar in most aspects, but the passive one is missing the motor design. While this is true, there is no question that a passive speaker is more expensive than a piece of plastic pipe (reference to bass reflex).
- Loudspeaker Design Cookbook 7th Edition by Vance Dickason (Audio Amateur Pubns, 2005).
- Audio Engineering Explained by Douglas Self (Taylor & Francis, 2012).
- Image source : link.
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